Skinny tires will have less rolling resistance over hard surfaces, and less sidewall deflection for better high-speed handling.
Wide tires have better floatation in mud or snow, but too wide, and they can get 'bouncy'

Farmr

 

Now theres an answer you can't argue with.

 

Generally people in the know will tell you to forgo the wider rear tires for any kind of low traction situation (i.e. snow, mud, etc.). Although some floatation is a good thing sometimes, the tires that have more PSI on the ground will get you home. Look at the ITP AG tire kit. No floatation there, but it will dig down and find terra firma when the "super swampers" are busy trying to spin with all of that weight and tire area. In areas where there are "high traction" 4x4 situations (rock climbing) the wider tires are the favorite due to their surface area.

On the other hand, some floatation is desirable if you get into a "quicksand" type mud pit. (one where there seems to be no bottom). Of course all of this is a moot point if you get your ATV stuck up to the frame. Nothing shy of a winch will help you there.